where and are the upwind static stability and horizontal wind speed respectively and so eq.(68) becomes:
This relation shows that vorticity is generated by two effects : changes in basic state density following an ascending or descending air parcel (first term on RHS of eq.(70) and baroclinic generation of vorticity on ascent or descent as represented by the second term. The density of a descending air parcel is never very different from its basic state value and so its volume decreases with time. One may liken the increase in vorticity caused by this compression to the increase in spin experienced by a rotating body when its moment of inertia is reduced (e.g. the spinning ice skater on pulling in his arms). The baroclinic term is harder to explain but encapsulates the net vorticity generation along a streamline due to the horizontal variation of the buoyancy force.
In a shallow layer of intense stability such as a boundary layer inversion, vertical displacement can lead to horizontal vorticity (and therefore vertical shear of the horizontal wind) of sufficient intensity for shearing
instability to develop.
A simple class of solution to eq.(70) is readily obtainable if one assumes constant U , B and \rho_0 . Defining the velocity streamfunction so that:
and
and substituting this into eq.(70) gives:
where . Now for the undisturbed flow and so eq.(71) may be written as: